Corollary 5

Let $\pi_1$ and $\pi_2$ be irreducible unitary representations of G. One has $\pi_1(\overline{\chi}_{\pi_2}) = 0$ if $\pi_1$ is not isomorphic to $\pi_2$, and $\pi_1(\overline{\chi}_{\pi_1})$ is multiplication by $1/$ dim($\pi_1$).

The character of $\pi_2$ is a continuous central function on G, hence the linear mapping $u$ = $\pi_1(\overline{\chi}_{\pi_2})$ is defined and belongs to the space Hom$_G(\pi_1, \pi_1)$. It is a homothety by Schur’s Lemma (prop. 6 of V, p. 386), whose trace is

Tr($u$) $=$ Tr$(\int_G\overline{\chi_{\pi_2}(g)}\pi_1(g)d\mu(g))$

By the orthogonality relations, the trace of $u$ is therefore zero if $\pi_1$ is not isomorphic to $\pi_2$, and equal to 1 otherwise. The assertion follows.